Study of Solutions to Some Functional Differential Equations with Piecewise Constant Arguments

Abstract

We provide optimal conditions for the existence and uniqueness of solutions to a nonlocal boundary value problem for a class of linear homogeneous second-order functional differential equations with piecewise constant arguments. The nonlocal boundary conditions include terms of the state function and the derivative of the state function. A similar nonhomogeneous problem is also discussed.

1. Introduction

In the study of second-order functional differential equations, there is a wide range of works dealing with periodic boundary value problems and piecewise continuous functional dependence. We mention, for instance, [1], where an equation independent of the first derivative is analyzed, and many other works as [2–14], where existence and stability results are provided.

Most of works on this field deal with nonconstructive existence results. However, in [7, 11], explicit solutions are found for second-order functional differential equations with piecewise constant arguments, through the calculus of the Green’s function. Other works in relation with first and higher-order differential equations with delay are [15–18].

In [19], a class of linear second-order differential equations with piecewise constant arguments is considered under the nonlocal conditions 𝑥(0)=𝜑, 𝑥(𝑇)=𝑥(𝜇)+𝜓, where 𝜑,𝜓∈ℝ, that is, an initial position is assumed and the boundary conditions are independent of the derivative of the function.

In this paper, we study the following nonlocal boundary value problem for homogeneous linear second-order functional differential equations:𝑥^′′(𝑡)+𝑎𝑥^′(𝑡)+𝑏𝑥(𝑡)+𝑐x^′𝑡𝑡[],𝑥(𝑇)𝜇𝑥([])+𝑑𝑥([])=0,𝑡∈𝐽=0,𝑇=𝑥+𝜑,_′(𝑇)=𝑥^′𝜇+𝜓, (1.1) for 𝑎,𝑏,𝑐,𝑑,𝜑,𝜓∈ℝ, 𝑇>0 and 𝜇∈(0,𝑇), where the functional dependence is given by the greatest integer part [𝑡], and the nonlocal boundary conditions involve both the state function and its derivative, which is the main difference from the study in [19]. A discussion for nonhomogeneous equations is also included. We consider the existence and uniqueness of solution to this problem, providing optimal conditions and calculating the exact expression of solutions.

To better illustrate the significant differences between the results included in this paper and those in [19], we remark that [19] is devoted to the study of the same class of linear second-order differential equations with piecewise constant arguments but considering a nonlocal boundary value problem where the value of the unknown function 𝑥 is fixed at the initial instant 𝑡=0 (i.e., an initial condition is imposed) and the boundary condition also involves the value of the sought solution at the right endpoint of the interval 𝑇 and an intermediate point 𝜇. None of the conditions imposed in [19] involve the value of the rate of change of the solution at any points. Thus, the nonlocal conditions in [19] can be reduced, by using the expression of the solution, to a boundary value problem affecting only the state of the solution, being independent of the rate of change of the state of the system. On the other hand, the nonlocal problem considered in this paper does not fix a certain initial position and, moreover, the boundary condition introduces the dependence, not only on the state, but also on the variation of the state of the system. Indeed, the value of the unknown function and its derivative is determined by the value of the corresponding magnitudes at an intermediate point of the interval of interest. Moreover, since the results in these two papers are also extensible to the special case where the intermediate point is identical to the initial instant (𝜇=0), we compare the consequences of both works to explain better the implications of the study of each problem. If we consider 𝜇=0, the results in [19] are applicable to obtain the solution to a class of linear second-order differential equations with piecewise constant arguments subject to the boundary conditions 𝑥(0)=𝜑, 𝑥(𝑇)=𝑥(0)+𝜓=𝜑+𝜓 and, therefore, we solve a problem with fixed values of the function at the end-points of the interval. However, again for 𝜇=0, the results presented in this paper allow to characterize the existence of solution (and provide its explicit expression) imposing boundary conditions of the type 𝑥(𝑇)=𝑥(0)+𝜑, 𝑥′(𝑇)=𝑥′(0)+𝜓, which include, as a particular case, periodic boundary conditions (on 𝑥 and 𝑥′).

The main results are stated after a few preliminary results are recalled, and, finally, examples are included to show the applicability of these results.

2. Preliminaries

Consider the equation𝑥^′′(𝑡)+𝑎𝑥^′(𝑡)+𝑏𝑥(t)+𝑐𝑥^′([𝑡])+𝑑𝑥([𝑡])=0,𝑡∈𝐽=[0,𝑇], (2.1) where 𝑎,𝑏,𝑐,𝑑 and 𝑇>0. The following concepts and results come from [7].

Definition 2.1. [7] Let the spaces {[𝑇,(𝑛Λ∶=𝑦∶𝐽⟶ℝ∶𝑦iscontinuousin𝐽⧵1,2,…,]}andthereexist𝑦⁻)∈ℝ,𝑦𝑛⁺(𝑛){[𝑇,𝐸=𝑦,∀𝑛∈1,2,…,]}∶=𝑥∶𝐽⟶ℝ∶𝑥,𝑥^′arecontinuousand𝑥^′′∈Λ. (2.2) A solution to (2.1) is a function 𝑥∈𝐸 which satisfies (2.1), taking 𝑥^′′(𝑛)=𝑥^′′(𝑛⁺),  forall𝑛∈{0,1,2,…,[𝑇]}.

For the constants 𝑎,𝑏,𝑐,𝑑∈ℝ, we define ℎ₁(𝑠) as 𝑑1−𝑎𝑑𝑠+𝑎²1−𝑒^−𝑎𝑠𝑑,if𝑏=0,𝑎≠0,1−2𝑠²𝑑,if𝑏=0,𝑎=0,1+𝑏𝑎1+2𝑠𝑒^{−(𝑎/2)𝑠}−𝑑𝑏,if𝑏≠0,𝑎²𝑑=4𝑏,1+𝑏𝛽𝑒^𝛼𝑠−𝛼𝑒^𝛽𝑠−𝑑𝛽−𝛼𝑏,ifb≠0,𝑎²𝑑>4𝑏,1+𝑏𝑒^{−(𝑎/2)𝑠}cos𝑎𝑏−²4𝑎𝑠+2𝑏−𝑎²/4sin𝑎𝑏−²4𝑠−𝑑𝑏,if𝑏≠0,𝑎²<4𝑏. (2.3) Consider also ℎ₂(𝑠) given by 1𝑎1−𝑒^−𝑎𝑠𝑐−𝑐𝑠+𝑎1−𝑒^−𝑎𝑠𝑐,if𝑏=0,𝑎≠0,𝑠−2𝑠²𝑒,if𝑏=0,𝑎=0,_{−(𝑎/2)𝑠}𝑐𝑏𝑎1+2𝑠−𝑐+𝑠𝑏,if𝑏≠0,𝑎²𝑒=4𝑏,𝛽𝑐/𝑏−1^𝛼𝑠+(1−𝛼𝑐/𝑏)𝑒^𝛽𝑠−𝑐𝛽−𝛼𝑏,if𝑏≠0,𝑎²𝑒>4𝑏,_{−(𝑎/2)𝑠}𝑐𝑏cos𝑎𝑏−²4()𝑠+1+𝑎𝑐/2𝑏𝑏−𝑎²/4sin𝑎𝑏−²4𝑠−𝑐𝑏,if𝑏≠0,𝑎²<4𝑏. (2.4) In the definitions of functions ℎ₁ and ℎ₂, we denote, for 𝑏≠0,𝑎²>4𝑏,𝑎𝛼=−2+𝑎2²𝑎−𝑏,𝛽=−2−𝑎2²−𝑏. (2.5)

It is easy to check [7] that ℎ₁(0)=1,ℎ₂(0)=0,ℎ^′₁(0)=0,ℎ^′₂(0)=1.

Theorem 2.2 (see [7, Theorem  2.1]). The initial value problem 𝑣^′′(𝑡)+𝑎𝑣^′(𝑡)+𝑏𝑣(𝑡)+𝑐𝑣^′𝑡𝑡[),𝑣(0)([])+𝑑𝑣([])=0,𝑡∈0,+∞=𝑣₀,𝑣^′(0)=𝑣^′₀, (2.6) for 𝑣₀,𝑣^′₀∈ℝ, has the solution 𝑣(𝑡)=ℎ₁(𝑡−𝑛)ℎ₂()𝐶𝑡−𝑛¹𝐶₂𝐶^′₁𝐶^′₂^𝑛𝑣₀𝑣^′₀,𝑡∈[𝑛,𝑛+1), (2.7) where 𝑛∈ℤ⁺, 𝐶₁=ℎ₁(1),𝐶₂=ℎ₂(1),𝐶^′₁=ℎ^′₁(1),𝐶^′₂=ℎ^′₂(1). (2.8)

To simplify calculus, for 𝑧∈[0,1], one denotes𝐻(𝑧)=ℎ₁(𝑧)ℎ₂(𝑧)ℎ^′₁(𝑧)ℎ^′₂(𝑧),(1)=𝐶𝐶=𝐻¹𝐶₂𝐶^′₁𝐶^′₂. (2.9)

3. Main Results

First, we consider the nonlocal boundary value problem𝑥^′′(𝑡)+𝑎𝑥^′(𝑡)+𝑏𝑥(𝑡)+𝑐𝑥^′𝑡𝑡[],𝑥(𝑇)𝜇𝑥([])+𝑑𝑥([])=0,𝑡∈𝐽=0,𝑇=𝑥+𝜑,_′(𝑇)=𝑥^′𝜇+𝜓, (3.1) for 𝑎,𝑏,𝑐,𝑑,𝜑,𝜓∈ℝ, 𝑇>0 and 𝜇∈(0,𝑇).

If 𝜇=0=𝜑, the condition 𝑥(𝑇)=𝑥(𝜇)+𝜑 is reduced to a periodic boundary condition of Dirichlet type. On the other hand, if 𝜇=0, we obtain 𝑥′(𝑇)=𝑥′(0)+𝜓, and, moreover, if 𝜓=0, we deduce the periodicity of the derivative of the function.

Theorem 3.1. If 𝑇>0 and 𝜇∈(0,𝑇), then problem (3.1) is solvable for each 𝜑𝜓 in the image of the mapping 𝑢𝑣ℱ∶⟶𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇𝐶^[𝜇]𝑢𝑣. (3.2) Hence, there exists a unique solution to (3.1) for every ^𝜑_𝜓∈ℝ² if and only if the matrix 𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇𝐶^[𝜇] (3.3) is nonsingular. In this case, the solution is given by 𝑣(𝑡)=ℎ₁(𝑡−𝑛)ℎ₂(𝑡−𝑛)𝐶^𝑛𝑣₀𝑣^′₀,𝑡∈[𝑛,𝑛+1),𝑛∈ℤ⁺, (3.4) where 𝑣₀𝑣^′₀ is the inverse image of 𝜑𝜓 by ℱ.
If the matrix (3.3) is singular, then there exists an infinite number of solutions to problem (3.1) for every 𝜑𝜓 in the image of ℱ, taking as initial position and initial slope in (3.4) any preimage of 𝜑𝜓 by ℱ, and there exist no solutions for the remaining pairs ^𝜑_𝜓∈ℝ².

Proof. We consider the initial value problem 𝑣^′′(𝑡)+𝑎𝑣^′(𝑡)+𝑏𝑣(𝑡)+𝑐𝑣^′𝑡𝑡[),𝑣(0)([])+𝑑𝑣([])=0,𝑡∈0,+∞=𝑣₀,𝑣^′(0)=𝑣^′₀, (3.5) for 𝑣₀,𝑣^′₀∈ℝ, whose solution is, by Theorem 2.2 [7, Theorem  2.1], 𝑣(𝑡)=ℎ₁(𝑡−𝑛)ℎ₂(𝑡−𝑛)𝐶^𝑛𝑣₀𝑣^′₀,𝑡∈[𝑛,𝑛+1), (3.6) where 𝑛∈ℤ⁺. We analyze under which conditions the boundary conditions are fulfilled, taking into account that 𝑣^′(𝑡)=ℎ^′₁(𝑡−𝑛)ℎ^′₂(𝑡−𝑛)𝐶^𝑛𝑣₀𝑣^′₀,𝑡∈[𝑛,𝑛+1), (3.7) where 𝑛∈ℤ⁺.
First, we consider the case 𝑇∉ℤ, and 𝜇∉ℤ. To obtain the solution to (3.1), we calculate 𝑣₀ and 𝑣^′₀ from the expressions of (3.6) and (3.7), in order to satisfy 𝑣(𝑇)=𝑣(𝜇)+𝜑 and 𝑣′(𝑇)=𝑣′(𝜇)+𝜓. Hence the boundary conditions are written as ℎ₁(𝑇−[𝑇])ℎ₂(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−ℎ₁𝜇−𝜇ℎ₂𝜇−𝜇𝐶^[𝜇]𝑣₀𝑣^′₀ℎ=𝜑,_′1(𝑇−[𝑇])ℎ^′₂(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−ℎ^′₁𝜇−𝜇ℎ^′₂𝜇−𝜇𝐶^[𝜇]𝑣₀𝑣^′₀=𝜓, (3.8) that is, 𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇𝐶^[𝜇]𝑣₀𝑣^′₀=𝜑𝜓. (3.9) The properties of functions ℎ₁ and ℎ₂ produce that the boundary conditions, in the cases where 𝑇∈ℤ, or 𝜇∈ℤ (or both), can be derived from the expression obtained in the case studied 𝑇∉ℤ, and 𝜇∉ℤ.
Therefore, this system has a solution only for 𝜑𝜓 in the image of the mapping 𝑢𝑣⟶𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇𝐶^[𝜇]𝑢𝑣. (3.10) Therefore, the solution is unique if the matrix (3.3) is nonsingular, in which case, the initial condition and initial slope are given as 𝑣₀𝑣^′₀=𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇𝐶^[𝜇]−1𝜑𝜓=𝐶^[𝜇]−1𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇⁻¹𝜑𝜓. (3.11) On the other hand, if the matrix (3.3) is singular, for 𝜑𝜓 in the image of ℱ, the infinitely many solutions are calculated from (3.6) taking as initial conditions any preimage of 𝜑𝜓 by ℱ, which proves the result.

Remark 3.2. The existence and uniqueness condition (3.3) is reduced, if [𝜇]>0, to the nonsingularity of the matrices 𝐶 and 𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇. (3.12) On the other hand, if [𝜇]=0, it is just reduced to the nonsingularity of the matrix [𝐻(𝑇−[𝑇])𝐶^[𝑇]−𝐻(𝜇)].

Remark 3.3. In Theorem 3.1, if we consider 𝜇=0, then the boundary conditions are 𝑥(𝑇)=𝑥(0)+𝜑,𝑥′(𝑇)=𝑥′(0)+𝜓, and the condition of existence and uniqueness of solution is reduced to the nonsingularity of 𝐻(𝑇−[𝑇])𝐶^[𝑇]−𝐼, (3.13) which coincides with condition (15) in [7, Theorem 2.2]. Moreover, if 𝜑=𝜓=0, the above-mentioned nonsingularity condition provides that the unique solution to the homogeneous equation subject to periodic boundary value conditions is the trivial solution (see [7, Theorem  2.2]).

Remark 3.4. In Theorem 3.1, the order 2 matrix (3.3) can be written in a simplified manner in the following particular cases.(i)If 𝑇∉ℤ and 𝑇−[𝑇]=𝜇−[𝜇], 𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐼𝐶^[𝜇]. (3.14)(ii)If 𝑇∉ℤ and 𝜇∈ℤ, 𝐻(𝑇−[𝑇])𝐶^{[𝑇]−𝜇}−𝐼𝐶^𝜇. (3.15)(iii)If 𝑇∈ℤ and 𝜇∉ℤ, 𝐶^{𝑇−[𝜇]}−𝐻𝜇−𝜇𝐶^[𝜇]. (3.16)(iv)If 𝑇,𝜇∈ℤ, 𝐶^𝑇−𝜇−𝐼𝐶^𝜇. (3.17)

Remark 3.5. Summarizing, the study of the solvability of (3.1) is reduced to the discussion of system (3.9). In the case of existence of an infinite number of solutions, we must analyze the rank of the matrix in (3.3) to determine whether any pair is admissible as initial position and slope or the space of initial conditions is one-dimensional. If the rank of (3.3) is zero, problem (3.1) is solvable only for ^𝜑_𝜓=⁰₀, that is, problem 𝑥^′′(𝑡)+𝑎𝑥^′(𝑡)+𝑏𝑥(𝑡)+𝑐𝑥^′𝑡𝑡[],𝑥(𝑇)𝜇,𝑥([])+𝑑𝑥([])=0,𝑡∈𝐽=0,𝑇=𝑥_′(𝑇)=𝑥^′𝜇, (3.18) for 𝑎,𝑏,𝑐,𝑑∈ℝ, 𝑇>0 and 𝜇∈(0,𝑇), has an infinite number of solutions, given by (3.4), for any initial point ^𝑣0_𝑣′0∈ℝ².

On the other hand, and denoting by 𝐺 the matrix in (3.3), if rank(𝐺)=1 and 𝜑𝜓 depends linearly on each column of 𝐺, then we have a one-dimensional space of solutions, whose starting conditions 𝑉₀=^𝑣0_𝑣^′₀ are determined from the row of the system 𝐺^𝑣0_𝑣^′₀=^𝜑_𝜓 corresponding to a nonzero minor of 𝐺.

Next, with the purpose of extending Theorem 3.1 to the nonhomogeneous case, we consider the following nonlocal boundary value problem for a nonhomogeneous equation:𝑥^′′(𝑡)+𝑎𝑥^′(𝑡)+𝑏𝑥(𝑡)+𝑐𝑥^′𝑡𝑡(𝑡)[],𝑥(𝑇)𝜇𝑥([])+𝑑𝑥([])=𝜎,𝑡∈𝐽=0,𝑇=𝑥+𝜑,_′(𝑇)=𝑥^′𝜇+𝜓, (3.19) for 𝑎,𝑏,𝑐,𝑑,𝜑,𝜓∈ℝ, 𝑇>0, 𝜇∈(0,𝑇), and 𝜎∈Λ. Using the expression of the solution for the corresponding initial value problem provided by [19, Theorem  5.2], we prove the following existence (and uniqueness) result.

Theorem 3.6. Consider 𝑎,𝑏,𝑐,𝑑∈ℝ, 𝑇>0, 𝜇∈(0,𝑇), 𝜎∈Λ and 𝜑,𝜓∈ℝ. Then problem (3.19) has a unique solution if and only if the matrix (3.3) is nonsingular. Under this assumption, the unique solution to (3.19) is given by 𝑥(𝑡)=ℎ₁(𝑡−𝑛)ℎ₂(𝑡−𝑛)𝐶^𝑛𝑉₀+^𝑛−1_𝑘=0𝑘^𝑘+1ℎ₁(𝑡−𝑛)ℎ₂(𝑡−𝑛)𝐶^{𝑛−1−𝑘}𝑔()𝑔𝑘+1−𝑠_′()𝜎(𝑠)+𝑘+1−𝑠𝑑𝑠_𝑡𝑛𝑔(𝑡−𝑠)𝜎(s)𝑑𝑠,𝑡∈[𝑛,𝑛+1),𝑛∈ℤ⁺, (3.20) with 𝑔 defined as 𝑔(𝑧)=1𝑎1−𝑒^−𝑎𝑧,if𝑏=0,𝑎≠0,𝑧,if𝑏=0,𝑎=0,𝑧𝑒^{−(𝑎/2)𝑧},if𝑏≠0,𝑎²𝑒=4𝑏,_𝛽𝑧−𝑒^𝛼𝑧𝛽−𝛼,if𝑏≠0,𝑎²1>4𝑏,𝑏−𝑎²𝑒/4^{−(𝑎/2)𝑧}sin𝑎𝑏−²4𝑧,if𝑏≠0,𝑎²<4𝑏, (3.21) and taking, as the initial condition 𝑉₀, 𝑉₀=𝐻(𝑇−[𝑇])𝐶^[𝑇]−𝐻𝜇−𝜇𝐶^[𝜇]−1𝜑𝜓−ℳ_{𝑇,𝜇,𝜎}, (3.22) where ℳ_{𝑇,𝜇,𝜎}=^[𝜇]−1_𝑘=0𝑘^𝑘+1𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇𝐶^{[𝜇]−1−𝑘}𝑔()𝑔𝑘+1−𝑠_′()𝜎(𝑠)+𝑘+1−𝑠𝑑𝑠_{[𝑇]−1𝜇𝑘=𝑘𝑘+1}𝐻(𝑇−[𝑇])𝐶^{[𝑇]−1−𝑘}𝑔()𝑔𝑘+1−𝑠_′()𝜎(𝑠)+𝑘+1−𝑠𝑑𝑠_𝑇[𝑇]𝑔()𝑔𝑇−𝑠_′()𝜎(𝑠)𝑇−𝑠𝑑𝑠−^𝜇_𝜇𝑔𝑔𝜇−𝑠_′𝜎(𝑠)𝜇−𝑠𝑑𝑠. (3.23) On the other hand, if the matrix (3.3) is singular, the number of solutions to (3.19) is determined by the discussion of the linear system 𝐻(𝑇−[𝑇])𝐶^[𝑇]−𝐻𝜇−𝜇𝐶^[𝜇]𝑉₀=𝜑𝜓−ℳ_{𝑇,𝜇,𝜎}, (3.24) and, for each solution 𝑉₀ to this system (in case it exists), the expression (3.20) provides a solution to (3.19).

Proof. The result follows from the expression of the solution (3.20) for the corresponding initial value problem, whose derivative is given by 𝑥′(𝑡)=ℎ^′₁(𝑡−𝑛)ℎ^′₂(𝑡−𝑛)𝐶^𝑛𝑉₀+^𝑛−1_𝑘=0𝑘^𝑘+1ℎ^′₁(𝑡−𝑛)ℎ^′₂(𝑡−𝑛)𝐶^{𝑛−1−𝑘}𝑔()𝑔𝑘+1−𝑠_′()𝜎(𝑠)+𝑘+1−𝑠𝑑𝑠_𝑡𝑛𝑔^′(𝑡−𝑠)𝜎(𝑠)𝑑𝑠,𝑡∈[𝑛,𝑛+1),𝑛∈ℤ⁺, (3.25) and the fact that the restrictions represented by the boundary conditions produce, respectively, 10𝐻(𝑇−[𝑇])𝐶^[𝑇]−𝐻𝜇−𝜇𝐶^[𝜇]𝑉₀=𝜑−10ℳ_{𝑇,𝜇,𝜎},01𝐻(𝑇−[𝑇])𝐶^[𝑇]−𝐻𝜇−𝜇𝐶^[𝜇]𝑉₀=𝜓−01ℳ_{𝑇,𝜇,𝜎}. (3.26)

4. Examples

We present some examples where different situations are analyzed in order to decide if the existence and uniqueness condition (nonsingularity of (3.3)) holds and, in case of nonuniqueness, the dimension of the space of solutions. The exact expression of the solution (or solutions) is given explicitly.

We recall that, for 𝑠∈[0,1], 𝐻(𝑠) is given, depending on the values of the coefficients 𝑎,𝑏,𝑐,𝑑 as follows.(i)If 𝑏=0,𝑎≠0, 𝐻(𝑠)=𝑑1−𝑎𝑑𝑠+𝑎²1−𝑒^−𝑎𝑠1𝑎1−𝑒^−𝑎𝑠𝑐−𝑐𝑠+𝑎1−𝑒^−𝑎𝑠𝑑𝑎−1+𝑒^−𝑎𝑠1𝑎(𝑎+𝑐)𝑒^−𝑎𝑠−𝑐, (4.1) thus (1)=𝐶𝐶=𝐻¹𝐶₂𝐶^′₁𝐶^′₂=𝑑1−𝑎+𝑑𝑎²1−𝑒^−𝑎1𝑎1−𝑒^−𝑎𝑐−𝑐+𝑎1−𝑒^−𝑎𝑑𝑎−1+𝑒^−𝑎1𝑎(𝑎+𝑐)𝑒^−𝑎−𝑐. (4.2) Note that the determinant of the matrix 𝐶 is equal to (𝐶)=𝑒det^−𝑎𝑒^𝑎−𝑎−1𝑑+𝑎−𝑎𝑒^𝑎𝑐+𝑎²𝑎². (4.3)(ii)If 𝑏=0, 𝑎=0, 𝐻(𝑠)=𝑑1−2𝑠²𝑐𝑠−2𝑠²−𝑑𝑠1−𝑐𝑠, (4.4) thus (1)=𝐶𝐶=𝐻¹𝐶₂𝐶^′₁𝐶^′₂=𝑑1−2𝑐1−2−𝑑1−𝑐. (4.5) The determinant of the matrix 𝐶 is equal to (𝐶)=𝑑det2−𝑐+1. (4.6)(iii)If 𝑏≠0,𝑎²=4𝑏, 𝐻(𝑠)=𝑑1+𝑏𝑎1+2𝑠𝑒^{−(𝑎/2)𝑠}−𝑑𝑏𝑒^{−(𝑎/2)𝑠}𝑐𝑏𝑎1+2𝑠−𝑐+𝑠𝑏𝑑1+𝑏−𝑎²4𝑠𝑒^{−(𝑎/2)𝑠}𝑒^{−(𝑎/2)𝑠}−𝑎²𝑐𝑎4𝑏𝑠−2𝑠+1, (4.7) thus (1)=𝑑𝐶=𝐻1+𝑏𝑎1+2𝑒^−𝑎/2−𝑑𝑏𝑒^−𝑎/2𝑐𝑏𝑎1+2−𝑐+1𝑏𝑑1+𝑏−𝑎²4𝑒^−𝑎/2𝑒^−𝑎/2−𝑎²𝑐−𝑎4𝑏2+1. (4.8) The determinant of the matrix 𝐶 is equal to det(𝐶)=𝑒^−𝑎(2𝑎−4)𝑒^𝑎/2+4𝑑−𝑎²𝑒^𝑎/2𝑐+4𝑏.4𝑏 (4.9)(iv)If 𝑏≠0,𝑎²>4𝑏, and denoting 𝑎𝛼=−2+𝑎2²𝑎−𝑏,𝛽=−2−𝑎2²𝐻(𝑠)=𝑑−𝑏,1+𝑏𝛽𝑒^𝛼𝑠−𝛼𝑒^𝛽𝑠−𝑑𝛽−𝛼𝑏𝛽𝑐/𝑏−1𝑒^𝛼𝑠+(1−𝛼𝑐/𝑏)𝑒^𝛽𝑠−𝑐𝛽−𝛼𝑏()𝑒𝑏+𝑑^𝛼𝑠−𝑒^𝛽𝑠()𝑒𝛽−𝛼𝑐−𝛼^𝛼𝑠+𝛽−𝑐𝑒^𝛽𝑠,𝛽−𝛼 (4.10) thus (1)=𝑑𝐶=𝐻1+𝑏𝛽𝑒^𝛼−𝛼𝑒^𝛽−𝑑𝛽−𝛼𝑏𝛽𝑐/𝑏−1𝑒^𝛼+(1−𝛼𝑐/𝑏)𝑒^𝛽−𝑐𝛽−𝛼𝑏𝑑1+𝑏𝑏𝑒^𝛼−𝑒^𝛽()𝑒𝛽−𝛼𝑐−𝛼^𝛼+𝛽−𝑐𝑒^𝛽𝛽−𝛼. (4.11) In this case, the determinant of the matrix 𝐶 is equal to (𝐶)=𝑑det1+𝑏𝛽𝑒^𝛼−𝛼𝑒^𝛽(𝑐−𝛼)𝑒^𝛼+𝛽−𝑐𝑒^𝛽𝛽−𝛼²−𝑒^𝛼−𝑒^𝛽𝛽𝑐−𝑏𝑒^𝛼+(𝑏−𝛼𝑐)𝑒^𝛽𝛽−𝛼²−𝑑𝑏(𝑐−𝛼)𝑒^𝛼+𝛽−𝑐𝑒^𝛽𝑑𝛽−𝛼+𝑐1+𝑏𝑒^𝛼−𝑒^𝛽=𝑑𝛽−𝛼1+𝑏𝛼²+𝛽²−2𝑏𝛽−𝛼²𝑒^𝛼+𝛽+1𝑑𝛽−𝛼𝑏𝛼𝑒^𝛼−𝑑𝑏𝛽𝑒^𝛽+𝑐𝑒^𝛼−𝑐𝑒^𝛽=𝑑1+𝑏𝑒^−𝑎+1𝑑𝛽−𝛼𝑏𝛼+𝑐𝑒^𝛼−𝑑𝑏𝛽+𝑐𝑒^𝛽. (4.12)(v)If 𝑏≠0,𝑎²<4𝑏, and denoting 𝑅=𝑏−𝑎²/4,𝐻(𝑠)=𝑑1+𝑏𝑒^{−(𝑎/2)𝑠}𝑎cos𝑅𝑠+−𝑑2𝑅sin𝑅𝑠𝑏𝑒^{−(𝑎/2)𝑠}𝑐𝑏()cos𝑅𝑠+1+𝑎𝑐/2𝑏𝑅−𝑐sin𝑅𝑠𝑏−1𝑅(𝑏+𝑑)𝑒^{−(𝑎/2)𝑠}sin𝑅𝑠𝑒^{−(𝑎/2)𝑠}cos𝑅𝑠−𝑎+2𝑐,2𝑅sin𝑅𝑠 (4.13)thus(1)=𝑑𝐶=𝐻1+𝑏𝑒^−𝑎/2𝑎cos𝑅+−𝑑2𝑅sin𝑅𝑏𝑒^−𝑎/2𝑐𝑏()cos𝑅+1+𝑎𝑐/2𝑏𝑅−𝑐sin𝑅𝑏−1𝑅(𝑏+𝑑)𝑒^−𝑎/2sin𝑅𝑒^−𝑎/2cos𝑅−𝑎+2𝑐.2𝑅sin𝑅 (4.14)The determinant of 𝐶 is given by (𝐶)=𝑑det1+𝑏𝑒^−𝑎cos²𝑐𝑅−𝑅𝑎()sin𝑅cos𝑅−𝑎+2𝑐4𝑅²sin²𝑅−𝑑𝑏𝑒^−𝑎/2𝑑cos𝑅+𝑏𝑎+2𝑐12𝑅sin𝑅+𝑅𝑑1+𝑏𝑐𝑒^−𝑎+sin𝑅cos𝑅𝑏+𝑑𝑅²𝑒^−𝑎1+𝑎𝑐2𝑏sin²𝑐𝑅−𝑅𝑑1+𝑏𝑒^−𝑎/2=𝑑sin𝑅1+𝑏𝑒^−𝑎cos²𝑑𝑅+1+𝑏𝑒^−𝑎−𝑎(𝑎+2𝑐)4𝑅²+𝑏𝑅²1+𝑎𝑐2𝑏sin²𝑅−𝑑𝑏𝑒^−𝑎/21cos𝑅+𝑅𝑑(𝑎+2𝑐)𝑑2𝑏−𝑐1+𝑏𝑒^−𝑎/2=𝑑sin𝑅1+𝑏𝑒^−𝑎−𝑑𝑏𝑒^−𝑎/21cos𝑅+𝑅𝑑(𝑎+2𝑐)𝑑2𝑏−𝑐1+𝑏𝑒^−𝑎/2sin𝑅. (4.15)

Example 4.1. Consider the problem 𝑥^′′(𝑡)+𝑥^′(𝑡)𝑡3+𝑑𝑥([])=0,𝑡∈𝐽=0,2,𝑥321=𝑥2𝑥+𝜑,_′32=𝑥^′12+𝜓, (4.16) where 𝑑,𝜑,𝜓∈ℝ. In this case, 𝑏=𝑐=0, 𝑎=1≠0, 𝑇=3/2 and 𝜇=1/2. Therefore, we get 𝐻(𝑠)=^{𝑑(1−𝑒−𝑠)−𝑑𝑠+11−𝑒−𝑠}_{𝑑𝑒−𝑠−𝑑𝑒−𝑠} so that matrix 𝐺=[𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻(𝜇−[𝜇])]𝐶^[𝜇]=𝐻(1/2)[𝐶−𝐼] is reduced to−𝑒𝐺=^−3/2√𝑒𝑑²+2𝑒𝑑−2𝑑²+(2−2𝑒)𝑑2𝑒^−3/2√𝑒(𝑒−1)𝑑+(2−2𝑒)𝑑+2𝑒−22𝑒^−3/2√𝑒𝑑²−𝑑²+(1−𝑒)𝑑−𝑒^−3/2√𝑒(𝑒−1)𝑑+(1−𝑒)𝑑+𝑒−1, (4.17)whose determinant is calculated as (𝐺)𝑒det=−⁻²√𝑒(3𝑒−3)𝑑²+(2−2𝑒)𝑑+2𝑒−2𝑒²𝑑²2𝑒=−^−3/2(3𝑒−3)+2𝑒⁻¹−2𝑑²+𝑒^−3/2(2−2𝑒)𝑑2. (4.18) Hence, 𝐺 is nonsingular if and only if 𝑑≠0 and 𝑑≠2/(3−2𝑒^1/2). In this case, problem (4.16) has a unique solution for every ^𝜑_𝜓∈ℝ², given by (see (3.4)) 𝑥(𝑡)=𝑑1−𝑒^−𝑡−𝑑𝑡+1𝑣₀+1−𝑒^−𝑡𝑣^′₀[),𝑥(𝑡)=,𝑡∈0,11−𝑒^1−𝑡𝑒⁻¹𝑑−𝑑𝑣₀+𝑒⁻¹𝑣^′₀+𝑑1−𝑒^1−𝑡−𝑑(𝑡−1)+11−𝑒⁻¹𝑑−𝑑+1𝑣₀+1−𝑒⁻¹𝑣^′₀=−𝑒^−𝑡−1𝑒𝑑−𝑑²𝑡+2𝑑²−𝑒𝑑−𝑒𝑒^𝑡−𝑒𝑑²+𝑒𝑑𝑣₀+((𝑒−1)𝑑𝑡+(2−2𝑒)𝑑−𝑒)𝑒^𝑡+𝑒²−𝑒𝑑+𝑒𝑣^′₀3,𝑡∈1,2, (4.19) where𝑣₀𝑣^′₀=𝐺⁻¹𝜑𝜓=−2√𝑒−1𝑑+22√𝑒−3𝑑²−√+2𝑑𝑒−2𝑑+22√𝑒−3𝑑²−2√+2𝑑𝑒−1𝑑−2𝑒+2√𝑒()+()−√2𝑒−23−3𝑒𝑑+2𝑒−2𝑒−2𝑑+2√𝑒𝑒−2𝑒+2√𝑒()+()𝜑𝜓,2𝑒−23−3𝑒𝑑+2𝑒−2 (4.20)that is, 𝑣₀=−2√𝑒−1𝑑+2𝜑−√𝑒−2𝑑+2𝜓2√𝑒−3𝑑²𝑣+2𝑑_′0=−2√𝑒−1𝑑−2𝑒+2𝜑−√𝑒−2𝑑+2√𝑒𝑒−2𝑒+2√𝑒()+().2𝑒−23−3𝑒𝑑+2𝑒−2 (4.21) For the particular case where 𝑑=2, 𝜑=1, and 𝜓=−1, then 𝑣₀𝑣^′₀=𝐺⁻¹1=−2√−1𝑒−14√−1𝑒−44−−𝑒+2√𝑒−1√𝑒()−2𝑒−2−2𝑒+2𝑒+11=−12𝑒−2−14√𝑒√1𝑒−12√𝑒√,𝑒−1 (4.22) and hence the unique solution to 𝑥^′′(𝑡)+𝑥^′(𝑡)𝑡3+2𝑥([])=0,𝑡∈𝐽=0,2,𝑥321=𝑥2𝑥+1,_′32=𝑥^′12−1, (4.23) is given by 𝑥(𝑡)=√𝑒(2𝑡−1)−2𝑒𝑡+𝑒−4𝑒+8√[),𝑥(𝑡)√𝑒−4,𝑡∈0,1=−𝑒2𝑒²+6𝑒𝑡−7𝑒²−21𝑒𝑒^𝑡+4𝑒³+12𝑒²4𝑒²+24𝑒+4𝑒^𝑡+1/2+−16𝑒²−16𝑒𝑒^𝑡−−6𝑒²−2𝑒𝑡+21𝑒²+7𝑒𝑒^𝑡−12𝑒³−4𝑒²4𝑒²+24𝑒+4𝑒^𝑡+1/2+−16𝑒²−16𝑒𝑒^𝑡3,𝑡∈1,2. (4.24) See Figure 1.
If 𝑑=0, then 𝐻(𝑠)=^{11−𝑒−𝑠}_0𝑒−𝑠 and the matrix 𝐺=𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇𝐶^[𝜇]1=𝐻2[𝐶−𝐼] (4.25) is reduced to 𝐺=^{0𝑒−3/2(𝑒−1)}_{0−𝑒−3/2(𝑒−1)}, which has rank one. Since 𝐺 is singular, there exists an infinite number of solutions to problem (4.16) for every 𝜑𝜓 in the image of the mapping given by the matrix 𝐺, that is, for every 𝜑−𝜑, where 𝜑∈ℝ. On each of these cases, the solutions are given by (3.4) taking as initial position and initial slope any preimage of 𝜑−𝜑, that is, the initial slope can be taken as 𝑣^′₀=𝑒^3/2𝜑,𝑒−1 (4.26) and the initial position can be chosen as any real number 𝑣₀.
Hence, the solutions to the problem 𝑥^′′(𝑡)(𝑡)3+𝑥′=0,𝑡∈𝐽=0,2,𝑥321=𝑥2𝑥+𝜑,_′32=𝑥^′12−𝜑, (4.27) where 𝜑∈ℝ, are given by 𝑥(𝑡)=𝑣₀+1−𝑒^−𝑡𝑣^′₀,𝑡∈[0,1), (4.28) and the same expression 𝑥(𝑡)=11−𝑒^{−(𝑡−1)}11−𝑒⁻¹0𝑒⁻¹𝑣₀𝑣^′₀=𝑣₀+1−𝑒^−𝑡𝑣^′₀3,𝑡∈1,2, (4.29) where 𝑣₀∈ℝ and 𝑣^′₀=𝑒^3/2𝜑/(𝑒−1). Hence, for each 𝜑 fixed, all the vertical shifts of the function 𝑥(𝑡)=(1−𝑒^−𝑡)(𝑒^3/2𝜑/(𝑒−1)),𝑡∈[0,3/2] are solutions to (4.27). In Figure 2, we show the graph of a solution which generates, by vertical shifting, all the solutions to problem (4.27) for 𝜑=1.
If 𝜓≠−𝜑, there is no solution for problem (4.16) with 𝑑=0.
Finally, if 𝑑=2/(3−2𝑒^1/2), then 𝐻(𝑠)=5−2√𝑒−2𝑠−2𝑒^−𝑠3−2√𝑒1−𝑒^−𝑠2𝑒^−𝑠−13−2√𝑒𝑒^−𝑠,1𝐺=𝐻2[]=√𝐶−𝐼4−2𝑒−2𝑒^−1/23−2√𝑒1−𝑒^−1/22𝑒^−1/2−13−2√𝑒𝑒^−1/2−2𝑒⁻¹3−2√𝑒1−𝑒⁻¹2𝑒⁻¹−13−2√𝑒𝑒⁻¹=−1−2𝑒^−3/2+2𝑒⁻¹+6𝑒^−1/2−10+4√𝑒3−2√𝑒²−𝑒^−3/2+𝑒⁻¹+𝑒^−1/2−13−2√𝑒2𝑒^−3/2−6𝑒^−1/2+43−2√𝑒²𝑒^−3/2−𝑒^−1/23−2√𝑒 (4.30) has rank one. Hence problem (4.16), for 𝑑=2/(3−2𝑒^1/2), has an infinite number of solutions for 𝜑𝜓 in the image of the mapping given by the matrix 𝐺, that is, for the values of 𝜑, and 𝜓 satisfying the system−2𝑒^−3/2+2𝑒⁻¹+6𝑒^−1/2−10+4√𝑒𝑣₀+−3𝑒^−3/2+5𝑒⁻¹+𝑒^−1/2−5+2𝑒^1/2𝑣^′₀=𝜑3−2√𝑒²,2𝑒^−3/2−6𝑒^−1/2+4𝑣₀+3𝑒^−3/2−3𝑒^−1/2−2𝑒⁻¹+2𝑣^′₀=𝜓3−2√𝑒², (4.31) for some values of 𝑣₀,𝑣^′₀∈ℝ. For the existence of these values of 𝑣₀,𝑣^′₀, it is necessary and sufficient that the rank of the matrix of the system (equal to one) coincides with the rank of the matrix −2𝑒^−3/2+2𝑒⁻¹+6𝑒^−1/2−10+4√𝑒−3𝑒^−3/2+5𝑒⁻¹+𝑒^−1/2−5+2𝑒^1/2𝜑3−2√𝑒²2𝑒^−3/2−6𝑒^−1/2+43𝑒^−3/2−3𝑒^−1/2−2𝑒⁻¹+2𝜓3−2√𝑒², (4.32) that is, for instance, det−3𝑒^−3/2+5𝑒⁻¹+𝑒^−1/2−5+2𝑒^1/2𝜑3𝑒^−3/2−3𝑒^−1/2−2𝑒⁻¹+2𝜓=0, (4.33) equivalent to 𝜓=3𝑒^−3/2−3𝑒^−1/2−2𝑒⁻¹+2−3𝑒^−3/2+5𝑒⁻¹+𝑒^−1/2−5+2𝑒^1/2𝜑. (4.34) Under this assumption, there exists an infinite number of solutions to problem 𝑥^′′(𝑡)+𝑥^′(𝑡)+23−2𝑒^1/2𝑥𝑡3([])=0,𝑡∈𝐽=0,2,𝑥321=𝑥2𝑥+𝜑,_′32=𝑥^′12+𝜓, (4.35) which are given by 𝑥(𝑡)=5−2√𝑒−2𝑡−2𝑒^−𝑡3−2√𝑒𝑣₀+1−𝑒^−𝑡𝑣^′₀[),𝑥(𝑡)=√,𝑡∈0,15−2𝑒−2(𝑡−1)−2𝑒^{−(𝑡−1)}3−2√𝑒1−𝑒^{−(𝑡−1)}3−2√𝑒−2𝑒⁻¹3−2√𝑒1−𝑒⁻¹2𝑒⁻¹−13−2√𝑒𝑒⁻¹𝑣₀𝑣^′₀=−2𝑒^−𝑡+4𝑒^1/2−𝑡+4√𝑒𝑡+4𝑒⁻¹𝑡−6𝑡+4𝑒−16√𝑒−8𝑒⁻¹+153−2√𝑒²𝑣₀+−𝑒^−𝑡−2𝑒^1−𝑡+2𝑒^1/2−𝑡+2𝑒⁻¹𝑡−2𝑡−2√𝑒−4𝑒⁻¹+73−2√𝑒𝑣^′₀3,𝑡∈1,2, (4.36) where 𝑣₀ and 𝑣^′₀ satisfy (one of) the equations in (4.31).
On the other hand, if (4.34) fails, there is no solution to (4.35).

Figure 1 

Solution to problem (4.23).

Figure 2 

Solution which generates by vertical shifts the remaining solutions to problem (4.27) with 𝜑=1.

Example 4.2. Next, consider the problem 𝑥^′′(𝑡)+𝑐𝑥^′𝑡𝑡3([])+𝑑𝑥([])=0,𝑡∈𝐽=0,2,𝑥321=𝑥2𝑥+𝜑,_′32=𝑥^′12+𝜓, (4.37) where 𝑐,𝑑,𝜑,𝜓∈ℝ. In this case, 𝑎=𝑏=0, 𝑇=3/2, and 𝜇=1/2. Hence, the matrix 𝐺=𝐻(𝑇−[𝑇])𝐶^{[𝑇]−[𝜇]}−𝐻𝜇−𝜇𝐶^[𝜇]1=𝐻2[𝐶−𝐼] (4.38) is reduced to 𝑑𝐺=²+(2𝑐−16)𝑑()16𝑐−2𝑑+2𝑐²−16𝑐+16𝑑16₂+(2𝑐−4)𝑑4(𝑐−2)𝑑+2𝑐²−4𝑐4, (4.39) whose determinant is det(𝐺)=𝑑((𝑑+8−4𝑐)/8). If 𝑑≠0 and 𝑑≠4𝑐−8, then 𝐺 is invertible and problem (4.37) has a unique solution for every ^𝜑_𝜓∈ℝ². Since 𝐺⁻¹=1()𝑑−4𝑐+82𝑐−4𝑑+4𝑐²−8𝑐𝑑−(𝑐−2)𝑑+2𝑐²−16𝑐+16−()2𝑑2𝑑+4𝑐−8𝑑+2𝑐−162, (4.40) the unique solution to (4.37), for 𝜑𝜓 fixed, is given by 𝑣(𝑡)=𝑑1−2𝑡²𝑣₀+𝑐𝑡−2𝑡²𝑣^′₀[),𝑣(𝑡)=𝑑,𝑡∈0,11−2(𝑡−1)²()−𝑐𝑡−12(𝑡−1)²𝑑1−2𝑐1−2𝑣−𝑑1−𝑐𝑣^0′₀=𝑑1−2(𝑡−1)²𝑑1−2+()−𝑐𝑡−12(𝑡−1)²(−𝑑)𝑣₀+𝑑1−2(𝑡−1)²𝑐1−2+()−𝑐𝑡−12(𝑡−1)²(1−𝑐)𝑣^′₀3,𝑡∈1,2, (4.41) where 𝑣₀=(2𝑐−4)𝑑+4𝑐²−8𝑐𝑑()()𝑑+8−4𝑐𝜑−𝑐−2𝑑+2𝑐²−16𝑐+16()𝑣2𝑑𝑑+8−4𝑐𝜓,_′0=−2𝑑+4𝑐−8𝑣𝑑−4𝑐+8⁰+𝑑+2𝑐−162()𝑣𝑑−4𝑐+8^′0. (4.42)
For instance, for 𝑐=2 and 𝑑=1, then 𝐻(𝑠)=^{1−𝑠2/2𝑠−𝑠2}_{−𝑠1−2𝑠} and 𝐺=^{−11/16−1/2}_1/40, which is invertible with inverse 𝐺⁻¹=⁰⁴_−2−11/2. Then the unique solution to (4.37), for 𝑐=2 and 𝑑=1 and 𝜑𝜓 fixed, is given by 𝑣(𝑡)=2𝑡²1−𝑡𝜑+27𝑡²[),𝑣(𝑡)1−11𝑡+8𝜓,𝑡∈0,1=−24𝑡²1−12𝑡+8𝜑+2−5𝑡²3+13𝑡−4𝜓,𝑡∈1,2. (4.43) See Figure 3 for the solution to (4.37) with 𝑐=2, 𝑑=1, 𝜑=1, and 𝜓=−1: 𝑣(𝑡)=12−3𝑡²[),1+7𝑡−8,𝑡∈0,12𝑡²3−𝑡−4,𝑡∈1,2. (4.44)
On the other hand, if 𝑑=0, then 𝐻(𝑠)=1𝑠−𝑐𝑠²2,101−𝑐𝑠𝐺=𝐻2[]=0𝑐𝐶−𝐼²−8𝑐+880𝑐²−2𝑐2 (4.45) is singular with rank 1, since 𝐺₁₂=0 if and only if 𝑐=4±2√2 and 𝐺₂₂=0 if and only if 𝑐=0 or 𝑐=2.
Hence, there exists an infinite number of solutions to problem (4.37) with 𝑑=0, for every 𝜑𝜓 in the image of the mapping given by the matrix 𝐺, that is, for every ^𝜑_𝜓∈ℝ² such that 4(𝑐²−2𝑐)𝜑=(𝑐²−8𝑐+8)𝜓. Under this assumption, the solutions to (4.37) with 𝑑=0 are given by 𝑣(𝑡)=𝑣₀+𝑐𝑡−2𝑡²𝑣^′₀[),𝑣(𝑡)=1()−𝑐,𝑡∈0,1𝑡−12(𝑡−1)²𝑐11−2𝑣01−𝑐𝑣^0′₀=𝑣₀+𝑐1−2+(𝑐1−𝑐)𝑡−1−2(𝑡−1)²𝑣^′₀3,𝑡∈1,2, (4.46) where the initial position 𝑣₀ can be chosen as any real number and the initial slope must satisfy 𝑣^′₀=8𝜑𝑐²√−8𝑐+8,if𝑐≠4±2𝑣2,_′0=2𝜓𝑐²−2𝑐,if𝑐≠0,𝑐≠2. (4.47)
For instance, if 𝑑=0 and 𝑐=1, then 𝐻(𝑠)=^{1𝑠−𝑠2/2}_01−𝑠, so that 𝐺=^01/8_0−1/2. Therefore, there exists an infinite number of solutions to problem (4.37) with 𝑑=0 and 𝑐=1, for every 𝜑𝜓 such that 𝜓=−4𝜑. In this case, the solutions to (4.37) with 𝑑=0 and 𝑐=1 are given by 𝑣(𝑡)=𝑣₀+81𝑡−2𝑡²[),3𝜑,𝑡∈0,14𝜑,𝑡∈1,2, (4.48) where 𝑣₀ is any real number. Note that 𝑣^′₀=8𝜑. All the solutions are obtained as vertical shifts of the function 𝑓(𝑡)=81𝑡−2𝑡²[),3𝜑,𝑡∈0,14𝜑,𝑡∈1,2. (4.49)
See Figure 4 for the graph of function 𝑓 for 𝜑=1, which generates by vertical shifting the one-dimensional space of solutions to problem (4.37) with 𝑑=0, 𝑐=1, 𝜑=1, and 𝜓=−4.
Finally, if 𝑑=4𝑐−8, then 𝐻(𝑠)=1−(2𝑐−4)𝑠²𝑐𝑠−2𝑠²(),−4𝑐+8𝑠1−𝑐𝑠𝐺=3𝑐²−18𝑐+2423𝑐²−16𝑐+1686𝑐²−24c+243𝑐²−10𝑐+82, (4.50) which has rank 1. Note that 𝐺₁₁≠0 for 𝑐≠2 and 𝑐≠4; 𝐺₁₂≠0 for 𝑐≠4/3 and 𝑐≠4; 𝐺₂₁≠0 for 𝑐≠2; and 𝐺₂₂≠0 for 𝑐≠2 and 𝑐≠4/3.
Therefore, problem (4.37) for 𝑑=4𝑐−8 has solution (an infinite number of solutions) if and only if 3𝑐²−18𝑐+242𝜓=6𝑐²−24𝑐+24𝜑,3𝑐²−16𝑐+168𝜓=3𝑐²−10𝑐+82𝜑, (4.51)(4.52) in which case the expressions of the solutions are given by (4.41), replacing the value of 𝑑, where 𝑣₀ and 𝑣^′₀ satisfy 3𝑐²−18𝑐+242𝑣₀+3𝑐²−16𝑐+168𝑣^′₀=𝜑,6𝑐²−24𝑐+24𝑣₀+3𝑐²−10𝑐+82𝑣^′₀=𝜓. (4.53)(4.54)
Indeed,(i)if 𝑐=2, then 𝐺=^0−1/2₀₀, and the problem has an infinite number of solutions for 𝜓=0 (see (4.52) since (4.51) is trivially satisfied), where 𝑣₀∈ℝ and 𝑣^′₀=−2𝜑 (from (4.53) since (4.54) is trivially satisfied),(ii)if 𝑐=4/3, then 𝐺=^8/30_8/30, and the problem has an infinite number of solutions for 𝜑=𝜓 (see (4.51) since (4.52) is trivially satisfied), where 𝑣₀=(3/8)𝜑=(3/8)𝜓 (from (4.53) or (4.54)) and 𝑣^′₀∈ℝ,(iii)if 𝑐≠2 and 𝑐≠4/3, then the elements in the second row of 𝐺 are nonzero and, hence, the problem has an infinite number of solutions for 𝜑=((3𝑐²−18𝑐+24)/(2(6𝑐²−24𝑐+24)))𝜓 (from (4.51) or the same expression 𝜑=((3𝑐²−16𝑐+16)/(4(3𝑐²−10𝑐+8)))𝜓 from (4.52)). The initial position and slope are taken satisfying (4.54). If we take 𝑐=4 (see the last case distinguished), then 𝐺=⁰⁰₂₄₁₆, and conditions (4.51) and (4.52) are reduced to 𝜑=0 and (4.54) is written as 24𝑣₀+16𝑣^′₀=𝜓.

Figure 3 

Solution to (4.37) for 𝑐=2, 𝑑=1, 𝜑=1, and 𝜓=−1.

Figure 4 

Solution which generates by vertical shifting the remaining solutions to problem (4.37) with 𝑑=0, 𝑐=1, 𝜑=1, and 𝜓=−4.

5. Conclusions

In this work, we have analyzed the existence and uniqueness of solutions to the nonlocal boundary value problem for linear second-order functional differential equations with piecewise constant arguments:𝑥^′′(𝑡)+𝑎𝑥^′(𝑡)+𝑏𝑥(𝑡)+𝑐𝑥^′𝑡𝑡(𝑡)[],𝑥(𝑇)𝜇𝑥([])+𝑑𝑥([])=𝜎,𝑡∈𝐽=0,𝑇=𝑥+𝜑,_′(𝑇)=𝑥^′𝜇+𝜓, (5.1) where 𝑎,𝑏,𝑐,𝑑,𝜑,𝜓∈ℝ, 𝑇>0, 𝜇∈(0,𝑇) and 𝜎∈Λ, by reducing the study to the discussion of a linear system of order 2. The nonlocal boundary conditions considered involve terms of the state function and the derivative of the state function, and the problem of interest also includes, as a particular case, periodic boundary value problems for second-order linear differential equations with functional dependence given by the greatest integer part.

Besides providing conditions on the existence and uniqueness of solutions, this approach also allows to obtain the expression of solutions explicitly, process which is illustrated with some examples.

The differences between the results in this paper and those in [19] have been included in the introduction.

Acknowledgments

The authors thank the editor and the anonymous referees for their helpful suggestions towards the improvement of the paper. This research is partially supported by Ministerio de Ciencia e Innovación and FEDER, Project MTM2010-15314.